The required values for m will be those that satisfy all the three constraints above. We can find those values by drawing the solution sets corresponding to the three inequalities on a real line and then determining their intersections:

We see that the intersection (common part) for the three sets is

\(m \ge 9\)

or \(m \in [9,\infty )\)

(d) We require at least one of the roots to be positive. This means that both the roots cannot be negative.

Hence, what we can do is first evaluate the values of m for which both roots are negative (or one root is zero and the other is negative), and then exclude these values from the set \(\left( { - \infty ,1]\, \cup \,[9,\infty } \right)\) {the set for which the roots are real}

To evaluate the set A (the values of m which both the roots are negative),

By now, you must have got the general idea about how to find constraints on the coefficients given some condition(s) on the roots. You need not remember all the cases separately. Just understand how they were derived; by using their corresponding graphs which immediately gave us the required conditions.

Example - 22

Find the values of a for which the inequality \({x^2} + ax + {a^2} + 6a < 0\) is satisfied for all \(x \in \left[ {1,2} \right]\)

Solution: Let us first graphically try to visualise what the problem statement given above means. Since \(f\left( x \right) = {x^2} + ax + {a^2} + 6a < 0\) for all \(x \in \left[ {1,2} \right],\) this means that the parabola for \(f\left( x \right)\) should remain below the x-axis for the entire interval [1, 2]; this can only happen if the interval [1, 2] falls between the roots of \(f\left( x \right):\)

This corresponds to “k1 and k2 lie between the roots”. The constraints are:

Solution: The coefficients of \({x^2}\) in both \(f\left( x \right)\) and \(g\left( x \right)\) are positive (both coefficients are 1) so the parabolas for both \(f\left( x \right)\,\,{\rm{and}}\,\,g\left( x \right)\) open upwards.

Since the roots of \(f\left( x \right)\) lie between the roots of \(g\left( x \right),\) the parabola for \(f\left( x \right)\) should lie “inside” the parabola for \(g\left( x \right);\) observe the figure below:

We can now say that since \(\alpha ,\beta \) lie between the roots of \(g\left( x \right),\)